Adapted Machine Repair Model

The simulated model is an adaptation of the Machine Repair Model and involves a truck that is sent for loading at a shovel or dumping (which is repair at a workshop in the original model) at every cycle with the number of shovels loading sides or tipping bins (repair bays) equal to R. The inter-arrival time at the shovel (travel full and empty, waiting time and dumping time) and the service time (loading time) are both assumed to have exponential distributions.

When arrivals to a system (trucks) are drawn from a small population the arrival rate may depend on the state of the system. The state of a system can be described as stable or unstable. Winston (2004) explains this concept by starting to define ρ as the traffic intensity for an M/M/1/GD/∞/∞ queue or model (recall such system will have exponential inter-arrival and service times).

Where, µ

ρ = λ (3.11)

λ = arrival rate, e.g. 5 trucks per hour arriving at a shovel.

µ = service rate or successful services per time unit e.g. a loading unit can service 10 trucks per hour.

Then for this example 0.5 10

5 =

ρ = <1 which means that the loader can handle the number of trucks dispatched to it. This will be explained in more detail in this section.

Winston (2004) also showed that an M/M/1/GD/∞/∞ queue process can be modelled as a birth-death process with the following parameters:

λ

λj = (j= 0,1,2,…)

0 =0

µ ^(3.12)

µ

µ_j = (j=1,2,3,…)

These equations are also called the flow balance equations of a birth-death process, with:

expected no. of departures from state j per unit time

= expected no. of entrances to state j per unit time

Winston then continued to define the steady state probabilities that j customers will be present as:

µ

π = λπ⁰, ₂⁰

2 2 λµπ

π = , …, _j

j

j λµπ

π = ⁰ (3.13)

πj is described as the probability that at an instant in the distant future, j customers (trucks) will be present or may be thought of (for the time in the distant future) as the fraction of time that the j customers are present at the shovel or dump.

Winston then defines:

1 ) 1

( +ρ+ρ² +⋅ ⋅⋅ =

π (3.14)

This infinite sum will diverge to infinity should ρ ≥1, and should ρ=1 no steady state will exist.

For the Machine Repair Model with a finite truck population it was stated that the arrival rate will depend on the state of the system. For example, should all the trucks

within a particular circuit be present at the loading unit and experiencing an unexpected breakdown; then the arrival rate will be zero. While at any other instant when there is less than the maximum number of trucks at the loading unit then the arrival rate will be positive.

It was stated in Section 3.1, that the Machine Repair Model can be described as M/M/R/GD/K/K, with inter-arrival and service times both having an exponential distribution, with R repair bays, K-trucks serviced by some general queue discipline and with the 2^nd K stating that the trucks come from a population with size K.

The length of time that a truck remains in good condition follows an exponential distribution with rateλ, and the time it takes to repair a truck can thus assumed to be exponential with rate µ .

The Machine Repair Model can be used to simulate shovel-truck load-and-haul cycles as indicated by the analogy in Table 3.4. It is the opinion of the author that this model has never been formally stated to be a valid and accurate calculation method for load-and-haul systems.

Table 3.4 Analogy between Machine Repair Model and Adapted Model for load-and-haul shovel- truck systems

Machine Repair Model Repair Model Adapted for Load and Haul System

L Expected number of broken trucks

Expected number of trucks at the loading unit or destination server (plant or dump)

Expected number of trucks waiting for service at the Workshop Repair Bays

Expected number of trucks waiting for service at the Loading unit or dumping destination

W Average time a machine spends broken (down time)

Average time a truck spends at the loading unit or dump destination.

Average time a truck spends waiting for service

Average time a truck queue at the loading unit or the plant/ dump.

Description Notation

L q

W q

The following section discusses how the model can be adopted to simulate shovel-truck load-and-haul systems

If we define µ

ρ = λ , the steady state probability distribution will be:

( )

and with (3.16) substituted in (3.15) which is expanded to include the K ^th state,

( )

To obtain W and W_q we will use Little’s Queuing Formula (Winston, 2004, p. 1226):

W

L =λ (3.20)

q

q W

L =λ (3.21)

The average number of truck arrivals per unit time is given byλ^{, where}

)

If (3.21) is applied to trucks being repaired (serviced by loading unit or dumping) and those trucks awaiting repairs (to be serviced), we obtain:

λ

W = L (3.23)

And applying (3.22) to trucks waiting to be repaired (to be serviced), we obtain:

λ

q q

W = L (3.24)

Later it will be shown that this model yields the same results when simulated in Excel compared to when simulated in Arena for a system with an exponential inter-arrival and service time.

Main observations made from this model are as follows:

• The Adapted Machine Repair Model can be programmed into Excel spreadsheets.

This gives it the ability to be linked with risk calculation packages such as

@Risk® program to provide a spread of results and to determine system sensitivities that can be used to plan risk aversive strategies.

• The benefit of using this model to simulate load-and-haul cycles is that it is stable in programs like Excel®.

• It can be expanded to include truck repair cycles.

• A further benefit is that the exact number of tipping bins (R) can be simulated to match real life mine sites and thus be used to motivate additional tipping bins or evaluate the impact of not having the sufficient number of bins.

• In general, engineers use spreadsheets to model aspects of projects and this model can easily interlink with these spreadsheets to deliver results faster.